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SOLUTIONSMANUAL TO ACCOMPANY ATKINS' PHYSICAL CHEMISTRY 397
�e fact that the vibrational frequency is lower in the ion than in the
neutral molecule is a result of the removal of a bonding electron: the ion
is less tightly held, and so has lower vibrational frequencies.
P11F.8 �e energy levels of a particle in a one-dimensional box are En = n2ħ2π2/2mL2.
�e linear triene has 6 π electrons which will occupy levels 1, 2 and 3 (pairwise),
so the HOMO–LUMO transition is from n = 3 to n = 4. If the average C–C
bond length is taken as d, then the triene is modelled as a box of length L = 5d.
�e transition energy is therefore
∆Elin = E4 − E3 =
(42 − 32)ħ2π2
2me(5d)2
= 7ħ2π2
50med2
�e energy levels of a particle on a ring of radius R are given by [7F.4–283],
Em = m2ħ2/2mR2, wherem = 0, ±1, ±2 . . ..�e cyclic triene has 6 π electrons
which will occupy levels 0, ±1 (pairwise), so the HOMO–LUMO transition is
from m = 1 to m = 2. If the average C–C bond length is taken as d, then the
radius of the circle into which a hexagon with this side can be inscribed is also
d.�e transition energy is therefore
∆Ecyc = E2 − E1 =
(22 − 12)ħ2
2med2
= 3ħ2
2med2
�e ratio between the transition energies is
∆Ecyc
∆Elin
= 3ħ2
2med2
× 50med
2
7ħ2π2
= 75
7π2
≈ 1.09
Ongoing from the linear to the cyclic polyene the energy gap therefore increases .
P11F.10 Tryptophan (Trp) and tyrosine (Tyr) show the characteristic absorption of a
phenyl group at about 280 nm. Cysteine (Cys) and glycine (Gly) lack the phenyl
group, as is evident from their spectra.
P11F.12 (a) �e ground state of O2 is a spin triplet, with Smol = 1. An oxygen atom
has the con�guration ... 2p4, which means that two of the electrons must
be paired up in a single p orbital, but that the other two can be arranged
to have parallel spins, S = 1, or paired spins S = 0.
According to theClebsch–Gordon series, [8C.6–333], combining Satom,1 =
1 with Satom,2 = 1 gives an overall spin of 2, 1, 0. �us, the two oxy-
gen atoms could both have S = 1 and combine to give Smol = 1 for the
molecule.
Similarly, combining Satom,1 = 1 with Satom,2 = 0 gives an overall spin of 1
only.�us, one of the oxygen atoms could have S = 1 and one could have
S = 0, and give Smol = 1 for the molecule.
�emolecule could dissociate into atomswhich are a singlet and a triplet
or into two triplets .